Optimal Triangulation

Years and Authors of Summarized Original Work

• 1972; Lawson

• 1992; Edelsbrunner, Tan, Waupotitsch

• 1993; Bern, Edelsbrunner, Eppstein, Mitchell, Tan

• 1993; Edelsbrunner, Tan

Problem Definition

Let S be a set of n points or vertices in $\mathbb{R}^{2}$. An edge is a closed line segment connecting two points. Let E be a collection of edges determined by vertices of S. The graph $\mathcal{G} = (S,E)$ is a plane geometric graph if (i) no edge contains a vertex other than its endpoints, that is, ab ∩ S = { a, b} for every edge ab ∈ E, and (ii) no two edges cross, that is, ab ∩ cd ∈ { a, b} for every two edges ab ≠ cd in E. A triangulation of S is a plane geometric graph $\mathcal{T} = (S,E)$ with E being maximal. Here maximality means that edges in E bound the convex hull of S, i.e., the smallest convex set in $\mathbb{R}^{2}$ that contains S, and subdivide its interior into disjoint faces bounded by triangles.

A plane geometric graph $\mathcal{G} = (S,E)$can be augmented with an...